cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A326522 Number of partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 16, 22, 25, 34, 39, 51, 58, 74, 84, 106, 118, 147, 165, 201, 223, 270, 301, 358, 396, 468, 518, 607, 666, 776, 853, 985, 1077, 1239, 1354, 1546, 1684, 1915, 2086, 2360, 2560, 2886, 3132, 3513, 3800, 4251
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2, where mu is the Möbius function (A008683).
a(n) = A326523(n)/n for n > 0.

A326523 Sum of all the parts in the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 10, 22, 24, 52, 70, 120, 144, 221, 288, 418, 500, 714, 858, 1173, 1392, 1850, 2184, 2862, 3304, 4263, 4950, 6231, 7136, 8910, 10234, 12530, 14256, 17316, 19684, 23673, 26640, 31816, 35826, 42355, 47388, 55755, 62284, 72662
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[n*Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Formula

a(n) = n * Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2, where mu is the Möbius function (A008683).
a(n) = n * A326522(n).
a(n) = A326524(n) + A326525(n) + A326526(n) + A326527(n) + A326528(n) + A326529(n) + A326530(n) + A326531(n) + A326532(n).

A326524 Sum of the smallest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 17, 23, 26, 36, 42, 55, 63, 80, 93, 119, 131, 165, 188, 230, 255, 312, 351, 420, 466, 555, 620, 731, 804, 945, 1046, 1216, 1333, 1550, 1702, 1959, 2141, 2452, 2688, 3064, 3334, 3790, 4136, 4673, 5070
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326525, A326526, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* Harvey P. Dale, Mar 22 2023 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * q, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).

A326526 Sum of the seventh largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 26, 31, 43, 51, 67, 78, 103, 119, 152, 172, 219, 250, 308, 348, 429, 486, 585, 658, 794, 892, 1063, 1185, 1410, 1572, 1847, 2053, 2407, 2670, 3095, 3425, 3964, 4380, 5030, 5532, 6344, 6974, 7939
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326527, A326528, A326529, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[All,7]]],{n,0,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 05 2020 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * o, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).

A326527 Sum of the sixth largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 16, 21, 29, 35, 49, 57, 77, 91, 118, 137, 177, 202, 255, 293, 363, 413, 509, 580, 707, 802, 969, 1097, 1319, 1481, 1764, 1980, 2337, 2615, 3069, 3421, 3982, 4431, 5126, 5689, 6553, 7240, 8301, 9169, 10451
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326528, A326529, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ]&][[All,6]]],{n,0,60}] (* Harvey P. Dale, Jul 05 2022 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * m, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).

A326528 Sum of the fifth largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 12, 18, 24, 34, 40, 56, 67, 91, 105, 138, 162, 209, 237, 304, 352, 441, 504, 630, 726, 893, 1016, 1236, 1409, 1700, 1912, 2287, 2579, 3052, 3417, 4018, 4492, 5237, 5824, 6756, 7508, 8655, 9561, 10967, 12114
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326529, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l * MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * l, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).

A326529 Sum of the fourth largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 22, 28, 40, 47, 67, 80, 107, 125, 167, 197, 257, 298, 387, 455, 575, 663, 834, 964, 1185, 1353, 1648, 1882, 2263, 2553, 3045, 3438, 4054, 4542, 5331, 5965, 6936, 7716, 8936, 9936, 11434, 12646, 14504
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326530, A326531, A326532.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k * MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * k, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326530(n) - A326531(n) - A326532(n).

A326530 Sum of the third largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 27, 34, 48, 57, 84, 100, 138, 166, 225, 269, 354, 416, 540, 633, 796, 920, 1153, 1324, 1616, 1845, 2238, 2546, 3056, 3445, 4109, 4640, 5471, 6139, 7231, 8100, 9453, 10560, 12291, 13710, 15870, 17622, 20327
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326531, A326532.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j * MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * j, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326531(n) - A326532(n).

A326531 Sum of the second largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 45, 70, 86, 124, 148, 207, 252, 334, 396, 520, 609, 781, 907, 1144, 1321, 1653, 1906, 2344, 2687, 3278, 3746, 4533, 5143, 6175, 6983, 8305, 9337, 11037, 12362, 14493, 16168, 18831, 20956, 24264, 26876
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326532.

Programs

  • Mathematica
    Join[{0},Table[Total[Select[IntegerPartitions[n,{9}],AllTrue[#,SquareFreeQ] &][[All,2]]],{n,60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 08 2020 *)

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * i, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326532(n).

A326532 Sum of the largest parts of the partitions of n into 9 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 34, 40, 60, 82, 124, 150, 216, 267, 371, 443, 595, 706, 941, 1090, 1423, 1656, 2110, 2415, 3026, 3485, 4286, 4876, 5937, 6749, 8131, 9145, 10932, 12308, 14580, 16302, 19190, 21446, 25053, 27865, 32372, 36004
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 11 2019

Keywords

Links

Crossrefs

Cf. A008683, A326522, A326523, A326524, A326525, A326526, A326527, A326528, A326529, A326530, A326531.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o-p-q) * MoebiusMu[q]^2 * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l - m - o - p - q]^2, {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 80}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} mu(q)^2 * mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p-q)^2 * (n-i-j-k-l-m-o-p-q), where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326524(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n).
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.